There was the Vacuum Oil Co. works in Wallasey Bridge Road, later Mobil, but it was not a refinery. Vacuum was associated with Standard Oil in America and had 2 ships which brought in the oils for blending into specialist lubricants for industry. It opened in 1901, with research laboratories included. It was an obvious target for bombing in WW2 and was rebuilt. It must have been an important plant, judging by the size of it along Wallasey Bridge Road.
Mobil went into petroleum refining and built the Coryton Refinery, now in the News
, in 1953, so presumably Birkenhead became less important and was closed in 2003.
It looks derelict now, on Streetview,
Another part of Birkenhead's important industrial past gone!!